Synchronous Grammars

Synchronous grammarsare a way of simultaneously generating

pairs of recursively related strings

Synchronous grammar

w wʹ

Synchronous grammars

Synchronous grammar

for i := 1 to 10 do begin n := n + iend

mov ax, 1loop: add bx, axcmp ax, 10jle loop

were originally invented for programming language compilation

Synchronous grammars

Synchronous grammar

I open the box open′(me′, box′)

have been proposed as a way of doing semantic interpretation

Synchronous grammars

Synchronous grammar

I open the box watashi wa hako wo akemasu

have been used for syntax-based machine translation

Synchronous grammarscan do much fancier transformations

than finite-state methods

shoonen ga gakusei ga sensei ga odotta to itta to hanashita

The boy stated that the student said that the teacher danced

boy student teacher danced statedsaid thatthat

Synchronous grammarscan do much fancier transformations

than finite-state methods

…dat Jan Piet de kinderen zag helpen zwemmen

…that John saw Peter help the children swim

John Peter the children saw swimhelp

Overview

Definitions

Properties

Algorithms

Extensions

Definitions

Context-free grammars

S→ NP VP

NP→ I

NP→ the box

VP→ V NP

V→ open

S

NP VP

V NPI

open the box

Context-free grammars

S→ NP VP

NP→ watashi wa

NP→ hako wo

VP→ NP V

V→ akemasu

S

NP VP

VNP

akemasuhako wo

watashi wa

Synchronous CFGs

S→ NP1 VP2 S→ NP1 VP2

NP→ I NP→ watashi wa

NP→ the box NP→ hako wo

VP→ V1 NP2 VP→ NP2 V1

V→ open V→ akemasu

Synchronous CFGs

S→ NP1 VP2, NP1 VP2

NP→ I, watashi wa

NP→ the box, hako wo

VP→ V1 NP2, NP2 V1

V→ open, akemasu

S→ NP1 VP2, NP1 VP2

NP→ I, watashi wa

NP→ the box, hako wo

VP→ V1 NP2, NP2 V1

V→ open, akemasu

NP2 VP3 NP2 VP3

S1 S1S1

VP3VP3

V4 NP5 V4NP5

NP2NP2

I watashi wa

Synchronous CFGs

V4V4

open akemasu

NP5NP5

the box hako wo

S1

S→ NP1 VP2, NP1 VP2

NP→ I, watashi wa

NP→ the box, hako wo

VP→ V1 NP2, NP2 V1

V→ open, akemasu

Adding probabilities0.3

0.1

0.6

0.5

0.2

NP2 VP3 NP2 VP3

S1 S1

0.3

Synchronous CFGs

S1 S1

Derivation probability:

V4V4

open akemasu

× 0.6

NP2NP2

I watashi wa

× 0.1

VP3VP3

V4 NP5 V4NP5

× 0.5

NP5NP5

the box hako wo

× 0.2

Other notations

(VP → V1 NP2 , VP → NP2 V1)

VP → 〈V NP〉

VP → ⋈[1,2] ( V NP )[2,1] V NP

Inversion transduction grammar (Wu)

Multitext grammar (Melamed)

VP → (V1 NP2 , NP2 V1)Syntax directed translation schema (Aho and Ullman;

Lewis and Stearns)

Properties

Chomsky normal form

X → Y Z

X → a

Chomsky normal form

A → (((B C) D) E) F rank 5

Chomsky normal form

A → [[[B C] D] E] F

A→ V1 F

V1→ V2 E

V2→ V3 D

V3→ B C

rank 5

rank 2

A hierarchy of synchronous CFGs

1-SCFG ⊊ 2-SCFG = 3-SCFG ⊊ 4-SCFG ⊊ …=ITG

(Wu, 1997)

=

1-CFG ⊊ 2-CFG = 3-CFG = 4-CFG = …

Synchronous CNF?

A → (B1 [C2 D3], [C2 D3] B1) rank 3

Synchronous CNF?

A → (B1 [C2 D3], [C2 D3] B1) rank 3

A→ (B1 V12 , V12 B1)

V1→ (C1 D2 , C1 D2)rank 2

Synchronous CNF?

A → (B1 C2 D3 E4, C2 E4 B1 D3)

A → ([B1 C2] D3 E4, [C2 E4 B1] D3)

A → (B1 [C2 D3] E4, [C2 E4 B1 D3])

A → (B1 C2 [D3 E4], C2 [E4 B1 D3])

rank 4

Synchronous CNF?

1 2 3 4

1 B2 C3 D4 E

1 2 3

1 B2 C3 D

A → (B1 C2 D3, C2 D3 B1)

A → (B1 C2 D3 E4, C2 E4 B1 D3)

A hierarchy of synchronous CFGs

1-SCFG ⊊ 2-SCFG = 3-SCFG ⊊ 4-SCFG ⊊ …=ITG

(Wu, 1997)

=

1-CFG ⊊ 2-CFG = 3-CFG = 4-CFG = …

Algorithms

Overview

Translation

Bitext parsing

I

Review: CKY

open the boxI

NP

V

open

S→ NP VPNP→ INP→ the boxVP→ V NP

V→ open

I

Review: CKY

open the boxI

NP

NP

the box

V

open

S→ NP VPNP→ INP→ the boxVP→ V NP

V→ open

V NP

I

Review: CKY

open the box

NP

NPV

VP

S→ NP VPNP→ INP→ the boxVP→ V NP

V→ open

NP VP

I

Review: CKY

open the box

NP VP

SS→ NP VPNP→ INP→ the boxVP→ V NP

V→ open

I

Review: CKY

open the box

SS→ NP VPNP→ INP→ the boxVP→ V NP

V→ open

Review: CKY

open the box

NP3V2

VP4

V2 NP3

O(n3) ways of matching

Translation

S

NP1 VP2

V3 NP4

open the box

I

I open the box

O(n3)

watashi wa

NP1

V3NP4

VP2

hako wo

NP4

akemasu

V3

Translation

S S

NP1 VP2

V3 NP4

open the box

I

I open the box

O(n3)

O(n)

NP1 VP2

S

watashi wa hako wo akemasu

O(n)

Translation

B C D E

A

What about…

O(n5) ways of combining?

Translation

B C D E

A

V2

V1B C D E

A

C E D B

Aflatten

O(n)translateO(n)

parseO(n3)

Bitext parsing

I open the box

S S

NP1 VP2 NP1 VP2

V3 NP4 V3NP4

open akemasuthe box hako wo

I watashi wa

O(n?)

watashi wa hako wo akemasu

V3 V3

open akemasu

NP4 NP4

the box hako wo

NP1 NP1

I watashi waopen akemasuthe box hako woI watashi wa

Bitext parsing

Consider rank-2 synchronous CFGs for now

V3 V3

open akemasu

NP4 NP4

the box hako wo

NP1 NP1

I watashi wa

Bitext parsing

VP2 VP2

V3 NP4 V3NP4

open akemasuthe box hako wo

NP1 NP1

I watashi wa

Bitext parsing

VP2 VP2

S S

NP1 VP2 NP1 VP2

open akemasuthe box hako woI watashi wa

Bitext parsing

S S

V3 V3

open akemasu

NP4 NP4

the box hako wo

Bitext parsing

VP2 VP2

V3 NP4 V3NP4

O(n6) ways of matching

Bitext parsing

B1 C2 D3 E4 C2 E4 B1 D3

A A

O(n10) ways of combining!

Algorithms

Translation: easy

Bitext parsing: polynomial in n but worst-case exponential in rank

Algorithms

Translation with an n-gram language model

Offline rescoring

Intersect grammar and LM (Wu 1996; Huang et al. 2005): slower

Hybrid approaches (Chiang 2005; Zollman and Venugopal 2006)

Extensions

Limitations of synchronous CFGs

S

NP VP

V NPJohn

misses Mary

S

NP VP

V

NP

Marie

manque

Jean

PP

P

à

One solution

S1

NP2 NP3

John

misses

Mary

S1

NP3 NP2

Marie

manque

Jean

à

Synchronous tree substitution grammars

S

NP1 VP

V NP2

John

misses

S

NP2 VP

V

NP1manque

PP

P

à

NP

Jean

NP

Mary

NP

Marie

NP

NPNP

Synchronous tree substitution grammarsS

VP

V NP

misses

S

VP

V

NPmanque

PP

P

à

NP

Mary

MarieNP

NP

John

Jean

NP

1

1

2

2

Limitations of synchronous TSGs

…dat Jan Piet de kinderen zag helpen zwemmen

…that John saw Peter help the children swim

This pattern extends to n nouns and n verbs

S

Limitations of synchronous TSGs

NPNP

S

VP

V

saw

S

VP

V

zag

S

NP

Jan

NP

John

1

2

1

2

Peter help the children swim

Piet de kinderen helpen zwemmen

? ?

Tree-adjoining grammarS

S V

zwemmen

de kinderen

NP VP

V

t

S

S V

helpen

Piet

NP VP

V

t

S*

S

S

V

zwemmen

de kinderen

NP VP

V

t

S

S V

helpen

Piet

NP VP

V

t

S

S V

zwemmen

de kinderen

NP VP

V

t

S

S V

helpen

Piet

NP VP

V

t

S*

Synchronous TAGS

S1

V

zwemmen

de kinderen

NP2

VP3

V

t

S1

the children

NP2

VP3

V

swim

S

S1

V

zwemmen

de kinderen

NP2

VP3

V

t

S1

the children

NP2

VP3

V

swim

S1

Peter

NP2

VP3

V

help

S*

S

S1

V

helpen

Piet

NP2

VP3

V

t

S*

S

the children

NP4

VP5

V

swim

S1

Peter

NP2

VP3

V

help

S

S

V

zwemmen

de kinderen

NP4

VP5

V

t

S

S1

V

helpen

Piet

NP2

VP3

V

t

S

the children

NP4

VP5

V

swim

S1

Peter

NP2

VP3

V

help

S

S

V

zwemmen

de kinderen

NP4

VP5

V

t

S

S1

V

helpen

Piet

NP2

VP3

V

t

S

S1

V

zag

Jan

NP2

VP3

V

t

S*

S1

John

NP2

VP3

V

saw

S*

S

the children

NP6

VP7

V

swim

S

Peter

NP4

VP5

V

help

S1

John

NP2

VP3

V

saw

S

S

V

zwemmen

de kinderen

NP6

VP7

V

t

S

S

V

helpen

Piet

NP4

VP5

V

t

S

V

zag

S1

Jan

NP2

VP3

V

t

Summary

Synchronous grammars are useful for various tasks including translation

Some rules “in the wild” (Chiang, 2005):X → (de, ’s)

X → (X1 de X2, the X2 of X1)

X → (X1 de X2, the X2 that)X → (zai X1 xia, under X1)

X → (zai X1 qian, before X1)X → (X1 zhiyi, one of X1)

Summary

Synchronous context-free grammars vary in power depending on rank

Translation is easy; bitext parsing is exponential in rank

Summary

Beyond synchronous CFGs,

synchronous TSGs allow multilevel rules

synchronous TAGs allow discontinuous constituents